where \(\operatorname{Ric}\)  is the Bochner operator on \(p\) -forms. Lott [1]} proved that if \(\operatorname{Ric}_f \ge 0\) , then all \(\Delta _f\) -harmonic 1-forms are parallel and, for compact manifolds, \(H^1(M;\mathbb {R})\)  is isomorphic to the space of all parallel 1-forms \(\omega \)  which satisfy \(\big \langle \nabla e^{-f}, \omega \big \rangle = 0\) . Moreover, if \(\operatorname{Ric}_f > 0\) , then all \(\Delta _f\) -harmonic 1-forms vanish.
